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# Evaluate the following logarithm

Evaluate:loge 2e3 + 2log3 - loge 18

### 2 Answers by Expert Tutors

Carl F. | Knowledgeable Chemistry tutorKnowledgeable Chemistry tutor
1.0 1.0 (1 lesson ratings) (1)
1
Properties of Logarithms:
logex = ln x
ln e = 1

a) logx(ab)= logx(a)+ logx(b)
b) logx(a/b)= logx(a) - logx(b)
c) logx(an) = n * logx (a)

so... log2e3 = (ln 2) + (3 * ln e) (properties a and c)
2loge3= ln (32) = ln 9 (reverse of property c)

ln 2 + 3*ln e + ln 9 = 3* ln (9*2*e) (combine the two and use property a )

3*ln 18e - ln 18 =3 ln (18e/18) = 3 ln e  (combine with 3rd eq, use property b and simplify)
since ln e = 1, 3 ln e = 3

Arthur D. | Effective Mathematics TutorEffective Mathematics Tutor
5.0 5.0 (11 lesson ratings) (11)
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ln(2e^3)+2ln(3)-ln(18)
ln(2)+ln(e^3)+2ln(3)-ln(18)
0.693147+3+2*(1.098612)-2.890371 (from tables)
0.693147+3+2.197224-2.890371
3.693147+2.197224-2.890371
5.890371-2.890371
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or ln(2)+ln(e^3)+ln(3)+ln(3)-ln(18)
ln(2)+ln(3)+ln(3)-ln(18)+ln(e^3)
ln(2*3*3)-ln(18)+ln(e^3)
ln(18)-ln(18)+ln(e^3)
ln(18/18)+ln(e^3)
ln(1)+ln(e^3)
0+ln(e^3)
ln(e^3)
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